Bulk Recombination Rate Constant

Fig. 16.2 Simplified kinetic model of the photocatalytic process. ps represents the light absorbed per unit surface area of the photocatalyst, e b and h+b are the photogenerated electrons and holes, respectively, in the semiconductor bulk, kR is the bulk recombination rate constant and /R the related flux, whatever recombination mechanism is operating A is the heat resulting from the recombination kDe and kDh are the net first-order diffusion constants for fluxes Je and Jh to the surface of e b and h+b in the semiconductor lattice, respectively e s and h+s are the species resulting from...

The breakdown of the diffusion theory of bulk ion recombination in high-mobility systems has been clearly demonstrated by the results of the computer simulations by Tachiya [39]. In his method, it was assumed that the electron motion may be described by the Smoluchowski equation only at distances from the cation, which are much larger than the electron mean free path. At shorter distances, individual trajectories of electrons were simulated, and the probability that an electron recombines with the positive ion before separating again to a large distance from the cation was determined. The value of the recombination rate constant was calculated by matching the net inward current of electrons... 

Here g is the Gartner flux of holes into the surface generated by illumination, [Red] is the bulk concentration of reduced species in solution and nxm.o is the electron density at x = 0. The rate constants, kET, for electron transfer involving free and trapped holes, and the recombination rate constant, kr, are shown in Fig. 8.5. They have units cm3 s-1 and can be converted to conventional heterogeneous rate constants (units cm4 s l) by multiplying by S. 

We saw that the bulk recombination rate was given by eqn. (351) where Cn and Cp are the trap capture rate constants (s 1) and nlr and ptr are defined above. At the surface, the relevant parameter of interest is the surface recombination rate, S, the number of recombining electron-hole pairs per unit area per second, which can be expressed [2] by... 

The value of the quantum yield of the primary charge separation products depends upon the ratio of the carriers separation and geminate recombination rate constants. The threshold (minimal energy) of the photoionization is determined by the energy necessary for an electron to leave any given molecule. As the photoionization takes place at the instant of the photon absorption the medium has no time to solvate the photoionization products (the electronic polarization of the medium, not the orientational, is important), so the effect of the medium upon the ionization threshold is relatively weak. For studies of the photoionization processes, the electron traps located in the bulk phase are usually used. 

Here, AE is the difference of the redox potentials of donor D and acceptor A, = 96500 C/mol q is the distribution coefficient of the radical ion which remains in the micellar phase, and q is the product of the distribution coefficients of the parent reactants A and D. If recombination can occur in the bulk phase or within the micellar phase but not at the interface, the value of recombination rate constant depends upon the minimal energy needed to bring one of the radical ions from the bulk to the micellar phase, i.e. fe = 10 q or from the micellar to the bulk phase with 10 / " dm /mole s (the highest of these values). So, the stabilization of the radical ions needs q to be small enough (less than 10" ) and q" to be high enough (more than 10 ). 

Fig. 10 KMC prediction of the bimolecular recombination rate constant symbols) for bulk heterojimctions with 4 nm (black) and 35 nm (redIgrey) domains. Squares correspond to no energetic disorder, while triangles assume Gaus sian-distributed energetic disorder with a = 15 meV. The solid line shows the prediction of the Langevin equation. Modified from [12]. American Physical Society...

Figure 13. Numerically calculated PMC potential curves from transport equations (14)—(17) without simplifications for different interfacial reaction rate constants for minority carriers (holes in n-type semiconductor) (a) PMC peak in depletion region. Bulk lifetime 10" s, combined interfacial rate constants (sr = sr + kr) inserted in drawing. Dark points, calculation from analytical formula (18). (b) PMC peak in accumulation region. Bulk lifetime 10 5s. The combined interfacial charge-transfer and recombination rate ranges from 10 (1), 100 (2), 103 (3), 3 x 103 (4), 104 (5), 3 x 104 (6) to 106 (7) cm s"1. The flatband potential is indicated.

The long-time limit of k(t) defines the steady state rate constant of the bulk ion recombination, which for the totally diffusion-controlled recombination is calculated as [27]... 

The rate constants given by Eqs. (35)-Eqs. (37) give the steady state rate constants of bulk ion recombination. As seen from Fig. 4, it takes some time before the steady state is established, and, in general, the rate coefficient of the bulk ion recombination is time-dependent. Although no simple analytical expression is available, which describes k t) in the whole time domain, the expression for the asymptotic behavior of k t) at long times is easily obtained. For fully diffusion-controlled recombination, we have... 

Figure 5 The rate constant of bulk electron-ion recombination, relative to the Debye-Smoluchowski value [Eq. (36)], as a function of the electron mean free path X. The solid line represents the simulation results, and the circles show the experimental data for liquid methane [49]. (From Ref. 39.)...

The above relationship between 0 and the rate constants is derived based on the conventional formulation of the rate equations. The unit to measure the amount of electrons and holes in the particle is density, the same as in bulk semiconductors. When the particle size is extremely small or the photon density is very low, only a few pairs of electron and hole are photogenerated and recombine with each other in the particle. This means that photon density does not take continuous values as suitably used in the conventional rate equations, but takes some series of values whose unit is the inverse of the particle volume. Taking into account this deviation, we proposed a new model in which particles are assigned by two integers, n and m, which represent the numbers of... 

The polyazophenylene units are formed from the polyrecombination of the decomposition products from bis(nitrosoacetyl)benzidine. Chain termination can occur by disproportionation of the polymer radicals and by recombination with acetoxy radicals. Despite the rate constant for the recombination of the phenyl and azophenyl radicals being much larger than that of the initiation reaction for isoprene, it is possible to synthesise copolymers from these materials by a careful choice of the various reaction parameters. However, block copolymers could only be obtained using emulsion techniques (see Table 4.11) and not in bulk or in solution. 

Solution of these equations eventually gives an expression fornj (0), the concentration of electrons at the surface, in terms of the parameters already defined, together with kCT, the rate constant for charge transfer across the interface (see Fig. 10.21) and two recombination constants, one for the bulk and one for the surface. Recombination of hole-electron pairs is taken into account in the development, as is also the formation of surface states by a surface-dependent anion adsorption at a degree of coverage, 9. 

Physically, kac and kK represent the rate constants for discharge of (a) the depletion layer charged by minority carriers that recombine through bulk or surface processes and (b) the faradaic transfer processes at the interface. 

As a rough rule of thumb, because the rate constants show little structural sensitivity, we can usually attribute the bulk of recombination to that radical which is present in highest concentration in the system. This can be determined generally from the propagation steps. 

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